Real and Complex Analysis (Rudin)

224 REAL AND COMPLEX ANALYSIS
The Calculus of Residues
10.41 Definition A function f is said to be meromorphic in an open set n if
there is a set A c n such that
(a) A has no limit point in n,
(b) fe H(n - A),
(c) fhas a pole at each point of A.
Note that the possibility A = 0 is not excluded. Thus every f e H(n) is
meromorphic in n.
Note also that (a) implies that no compact subset of n contains infinitely
many points of A, and that A is therefore at most countable.
Iffand A are as above, if a e A, and if
m
Q(z) = L Ck(Z - a)-k (1)
k=1
is the principal part off at a, as defined in Theorem 10.21 (i.e., iff - Q has a
removable singularity at a), then the number C1 is called the residue offat a:
C1 = Res (f; a).
If r is a cycle and a ¢ r*, (1) implies
~ r Q(z) dz = C1 Indr (a) = Res (Q; a) Indr (a).
2m Jr
This very special case of the following theorem will be used in its proof.
(2)
(3)
10.42 The Residue Theorem Suppose f is a meromorphic function in n. Let A
be the set of points in n at whichfhas poles. Ifr is a cycle in n - A such that
then
Indr (/X) = 0 for all (1)
-2 1 . r f(z) dz = L Res (f; a) Indr (a).
1t1 Jr
a E A
(2)
PROOF Let B = {a e A: Indr (a) =1= OJ. Let W be the complement of r*. Then
Indr (z) is constant in each component V of W. If- V is unbounded, or if V
intersects n c , (1) implies that Indr (z) = 0 for every z e V. Since A has no
limit point in n, we conclude that B is a finite set.
The sum in (2), though formally infinite, is therefore actually finite.
Let a1' ... , an be the points of B, let Qlo ... , Qn be the principal parts off
at alo ..., a., and put g = f - (Q1 + ... + Q.). (If B = 0, a possibility which
is not excluded, then g = f.) Put no = n - (A - B). Since g has removable